\[
\newcommand{\mY}{\mathbf{Y}}
\newcommand{\mZ}{\mathbf{Z}}
\newcommand{\mX}{\mathbf{X}}
\newcommand{\mU}{\mathbf{U}}
\newcommand{\mV}{\mathbf{V}}
\newcommand{\mD}{\mathbf{D}}
\newcommand{\vg}{\boldsymbol{\gamma}}
\newcommand{\vd}{\boldsymbol{\delta}}
\newcommand{\ybar}{\bar{y}}
\newcommand{\vgs}{\mathbf{g}}
\newcommand{\ves}{\mathbf{e}}
\newcommand{\va}{\mathbf{a}}
\newcommand{\vx}{\mathbf{x}}
\]
Abstract
The models behind the analyses and the notation of the help pages.
Notation
symbol text kind
<char> <char> <char>
1: G G scalar
2: E E scalar
3: B B scalar
4: i, j, k i j k index
5: \\mathbf{Y} = (\\bar{y}_{ij}) Y matrix
6: \\bar{y}_{i\\cdot}, \\bar{y}_{\\cdot j}, \\bar{y} y_i. y_.j y.. scalar
7: \\mathbf{Z} = (z_{ij}) Z matrix
8: I_j I_j scalar
9: s^2, f_e s2 f_e scalar
10: \\beta_i b_i scalar
11: S^2_{d_i} S2d_i scalar
12: W_i W_i scalar
13: \\sigma^2_i sigma2_i scalar
14: \\lambda_n lambda_n scalar
15: \\boldsymbol{\\gamma}_n, \\boldsymbol{\\delta}_n gamma_n delta_n vector
16: P P scalar
17: \\mathbf{g}_i, \\mathbf{e}_j g_i e_j vector
18: \\mathbf{a} a vector
19: \\alpha, Z_{1-\\alpha} alpha z scalar
20: P_i P_i scalar
21: S^{(1)}_i, S^{(2)}_i S1_i S2_i scalar
22: \\mathbf{A}^\\top, \\mathbf{1} A^T 1 operator
symbol text kind
<char> <char> <char>
meaning
<char>
1: number of genotypes
2: number of environments
3: number of replicates in each environment
4: genotype, environment, replicate
5: table of genotype-by-environment means
6: genotype, environment and grand means
7: interaction effects (means centred on rows and columns)
8: environment index, environment mean minus grand mean
9: pooled error mean square of the plots and its degrees of freedom
10: regression slope of genotype i on the environment index
11: Eberhart and Russell's deviation variance
12: Wricke's ecovalence
13: Shukla's stability variance
14: n-th singular value (AMMI: of Z; GGE: of the environment-centred means)
15: genotype and environment singular vectors
16: singular value partition of the GGE biplot: 1 genotype-focused, 0.5 symmetric, 0 environment-focused
17: GGE biplot points of genotype i and environment j
18: direction of the average environment axis
19: risk of a low yield accepted in the safety-first and reliability indices, and the normal quantile
20: Lin and Binns' superiority measure
21: Huehn's mean absolute rank difference and variance of ranks
22: transpose, vector of ones
meaning
<char>
argument
<char>
1: .gen
2: .env
3: .rep
4:
5: .y
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16: .svp
17:
18:
19: alpha
20:
21:
22:
argument
<char>
The two-way table
All methods start from the table of means \(\mY = (\ybar_{ij})\), \(G\) genotypes by \(E\) environments, each the mean of \(B\) replicates. The additive model and the interaction:
\[
\ybar_{ij} = \mu + g_i + e_j + z_{ij}, \qquad z_{ij} = \ybar_{ij} - \ybar_{i\cdot} - \ybar_{\cdot j} + \ybar .
\]
Regression methods model the interaction as a linear function of the environment index \(I_j = \ybar_{\cdot j} - \ybar\): \(\ybar_{ij} = \mu_i + \beta_i I_j + \delta_{ij}\).
Variance methods summarise each row of \(\mZ\): Wricke’s ecovalence \(W_i = \sum_j z_{ij}^2\) and Shukla’s stability variance \(\sigma^2_i = [G(G-1)W_i - \sum_k W_k]/[(E-1)(G-1)(G-2)]\).
AMMI decomposes \(\mZ\) by its singular values, \(\mZ = \sum_n \lambda_n \vg_n \vd_n^\top\), and tests each term.
GGE decomposes the environment-centred table \(\mX = (\ybar_{ij} - \ybar_{\cdot j})\), genotype main effect plus interaction, and plots its first two terms.
Scales
The analyses of variance are on the scale of the plots; the stability statistics on the scale of the means. A mean square of the means corresponds to a plot mean square divided by \(B\); the pooled error of the means is \(s^2/B\).
Summary
One table of means, four views of it: regression, variance, AMMI and GGE.
References
Becker, H. C. and Leon, J. (1988). Stability analysis in plant breeding. Plant Breeding, 101(1), 1–23.
Gauch, H. G. (1992). Statistical Analysis of Regional Yield Trials: AMMI Analysis of Factorial Designs. Elsevier, Amsterdam.
Yan, W., Hunt, L. A., Sheng, Q. and Szlavnics, Z. (2000). Cultivar evaluation and mega-environment investigation based on the GGE biplot. Crop Science, 40(3), 597–605.